Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints

Abdelmouhcene Sengouga

doi:10.3934/eect.2020014

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2020, Volume 9, Issue 1: 1-25. Doi: 10.3934/eect.2020014

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Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints

Abdelmouhcene Sengouga

Laboratory of Functional Analysis and Geometry of Spaces, Faculty of Mathematics and Computer Sciences, University of M'sila, 28000, Algeria

Received: October 31, 2017

Revised: April 30, 2018

Early access: October 2019

Published: March 2020

The author is supported by the Algerian Ministry of Higher Education and Scientific Research, under a PRFU Project No. C00L03UN280120180007

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Abstract

We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decays at the rate $ 1/t $. We also establish observability results, at one or at both endpoints, in a sharp time. Moreover, using the Hilbert uniqueness method, we derive exact boundary controllability results.

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Mathematics Subject Classification: 35L05, 93B05, 93B07.

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Figure 1. Extension of an initial data $ \phi ^{0} $ when $ v _{1}<v _{2} $

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Figure 2. Propagation of a wave with a small support near an endpoint $ (v _{2}<v _{1}) $

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Figure 3. Propagation of small disturbances with supports near one or two ends $ (v _{1}<v _{2}) $

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